1 Nonparametric Methods III Henry Horng-Shing Lu Institute of Statistics National Chiao Tung University [email protected]

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Nonparametric Methods III

Henry Horng-Shing LuInstitute of Statistics

National Chiao Tung [email protected]

http://tigpbp.iis.sinica.edu.tw/courses.htm

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PART 4: Bootstrap and Permutation Tests Introduction References Bootstrap Tests Permutation Tests Cross-validation Bootstrap Regression ANOVA

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References Efron, B.; Tibshirani, R. (1993). An Introduction

to the Bootstrap. Chapman & Hall/CRC. http://cran.r-project.org/doc/contrib/Fox-Co

mpanion/appendix-bootstrapping.pdf http://cran.r-project.org/bin/macosx/2.1/chec

k/bootstrap-check.ex http://bcs.whfreeman.com/ips5e/content/cat

_080/pdf/moore14.pdf

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Hypothesis Testing (1) A statistical hypothesis test is a method of m

aking statistical decisions from and about experimental data.

Null-hypothesis testing just answers the question of “how well the findings fit the possibility that chance factors alone might be responsible.”

This is done by asking and answering a hypothetical question.

http://en.wikipedia.org/wiki/Statistical_hypothesis_testing

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Hypothesis Testing (2) Hypothesis testing is largely the product of Ronald Fisher,

Jerzy Neyman, Karl Pearson and (son) Egon Pearson. Fisher was an agricultural statistician who emphasized rigorous experimental design and methods to extract a result from few samples assuming Gaussian distributions. Neyman (who teamed with the younger Pearson) emphasized mathematical rigor and methods to obtain more results from many samples and a wider range of distributions. Modern hypothesis testing is an (extended) hybrid of the Fisher vs. Neyman/Pearson formulation, methods and terminology developed in the early 20th century.

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Hypothesis Testing (3)

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Hypothesis Testing (7) Parametric Tests:

Nonparametric Tests: Bootstrap Tests Permutation Tests

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Confidence Intervals vs. Hypothesis Testing (1)

Interval estimation ("Confidence Intervals") and point estimation ("Hypothesis Testing") are two different ways of expressing the same information.

http://www.une.edu.au/WebStat/unit_materials/c5_inferential_statistics/confidence_interv_hypo.html

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If the exact p-value is reported, then the relationship between confidence intervals and hypothesis testing is very close. However, the objective of the two methods is different: Hypothesis testing relates to a single

conclusion of statistical significance vs. no statistical significance.

Confidence intervals provide a range of plausible values for your population.

http://www.nedarc.org/nedarc/analyzingData/advancedStatistics/convidenceVsHypothesis.html

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Which one? Use hypothesis testing when you want to do a

strict comparison with a pre-specified hypothesis and significance level.

Use confidence intervals to describe the magnitude of an effect (e.g., mean difference, odds ratio, etc.) or when you want to describe a single sample.

http://www.nedarc.org/nedarc/analyzingData/advancedStatistics/convidenceVsHypothesis.html

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P-value

http://bcs.whfreeman.com/ips5e/content/cat_080/pdf/moore14.pdf

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Achieved Significance Level (ASL)

*0 0

0

Definition:

The (ASL) is defined as:

ˆ ÂSL = ( | H ).

The smaller ASL, the stronger is the evidence of false.

Th

achieved significance

e ASL is an estimate o

l

f

e

the p-value by

vel o

perm

f the test

P

H

uation and bootstrap methods.

0

Definition

A is a way of deciding whether or not the data decisively

reject the h

hypoth

ypothe

esis

sis

test

.

H

https://www.cs.tcd.ie/Rozenn.Dahyot/453Bootstrap/05_Permutation.pdf

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Bootstrap Tests Methodology Flowchart R code

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Bootstrap Tests Beran (1988) showed that bootstrap inference

is refined when the quantity bootstrapped is asymptotically pivotal.

It is often used as a robust alternative to inference based on parametric assumptions.

http://socserv.mcmaster.ca/jfox/Books/Companion/appendix-bootstrapping.pdf

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Hypothesis Testing by a Pivot

2 1

2

1

:

- 1. : (0, 1), ( , ) , , .

-2. : = (0, 1) , ( , ),

( ,

n

ii

i

Di

n

ii

Examples

XX

A pivot Z N when X iid N and X is knownn

n

XAn asymptotic pivot T N as n when X iid N

Sn

Xwhere X is unknown and S

n

2

1

).

1

n

ii

X X

n

http://en.wikipedia.org/wiki/Pivotal_quantity

Pivot or pivotal quantity: a function of observations whose distribution does

not depend on unknown parameters.

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T statistics can be regarded as a pivot or an asymptotic pivotal when the data are normally distributed.

Bootstrap T tests can be applied when the data are not normally distributed.

One Sample Bootstrap Tests

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Bootstrap T tests Flowchart R code

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·0

1 2 0

ˆˆ ( , , ..., ) ( ),

ˆ( )ndata x x x x x s x and t

Bootstrap B times*Bx*2x*1x

*Bt

*2t

*1t

· *0#{ }/Boot bASL t t B

Flowchart of Bootstrap T Tests

** 0

*

ˆ

ˆ( )

bb

b

t

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Bootstrap T Tests by R

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An Example of Bootstrap T Tests by R

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Bootstrap Tests by The “BCa” The BCa percentile method is an efficient met

hod to generate bootstrap confidence intervals.

There is a correspondence between confidence intervals and hypothesis testing.

So, we can use the BCa percentile method to test whether H0 is true.

Example: use BCa to calculate p-value

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Use R package “boot.ci(boot)” Use R package “bcanon(bootstrap)” http://qualopt.eivd.ch/stats/?page=bootstrap http://www.stata.com/capabilities/boot.html

BCa Confidence Intervals:

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http://finzi.psych.upenn.edu/R/library/boot/DESCRIPTION

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An Example of “boot.ci(boot)” in R

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http://finzi.psych.upenn.edu/R/library/bootstrap/DESCRIPTION

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An example of “bcanon(bootstrap)” in R

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BCa by http://qualopt.eivd.ch/stats/?page=bootstrap

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Use BCa to calculate p-value by R

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Two Sample Bootstrap Tests Flowchart R code

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Bootstrap B times* * *1 1 1( , )d y x

1 2 1: ( , y , ..., y )nSample yy

* * *1 1 1ˆ ( ) ( )s s y x

· *ˆ ˆ (#( )) /Boot bASL B

Flowchart of Two-Sample Bootstrap Tests

1 2 2: ( , x , ..., x )mSample xx

1 2 1ˆ : ( , , ..., , , ..., ) ( , ) ( ) ( )n n n mcombined data d d d d d s s d y x y x

* * *2 2 2( , )d y x

* * *2 2 2

ˆ ( ) ( )s s y x * * *ˆ ( ) ( )B B Bs s y x

m+n=Ncombine

* * *( , )B B Bd y x

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Two-Sample Bootstrap Tests by R

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Output (1)

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Output (2)

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Permutation Tests Methodology Flowchart R code

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Permutation In several fields of mathematics, the term

permutation is used with different but closely related meanings. They all relate to the notion of (re-)arranging elements from a given finite set into a sequence.

http://en.wikipedia.org/wiki/Permutation

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Permutation Tests Permutation test is also called a

randomization test, re-randomization test, or an exact test.

If the labels are exchangeable under the null hypothesis, then the resulting tests yield exact significance levels.

Confidence intervals can then be derived from the tests.

The theory has evolved from the works of R.A. Fisher and E.J.G. Pitman in the 1930s.

http://en.wikipedia.org/wiki/Pitman_permutation_test

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Applications of Permutation Tests (1)

We can use a permutation test only when we can see how to resample in a way that is consistent with the study design and with the null hypothesis.


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Two-sample problems when the null hypothesis says that the two populations are identical. We may wish to compare population means, proportions, standard deviations, or other statistics.

Matched pairs designs when the null hypothesis says that there are only random differences within pairs. A variety of comparisons is again possible.

Relationships between two quantitative variables when the null hypothesis says that the variables are not related. The correlation is the most common measure of association, but not the only one.


Applications of Permutation Tests (2)

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2

2a b

0

A tradionnal way is to consider some hypotheses: ~ ( , )

and ~ ( , ), and the null hypothesis becomes = .

Ûnder , the statistic . = - can be modelled as a normal

distribution with mean

a

b

a b

F N

F N

H X X

* 2

2

2 2ˆ

ˆ ˆ( )

2*

ˆˆ

2 2

2 1 1

1 1 0 and variance ( ).

The ASL is then computed by

ˆ ASL=2

when is unknown and has to be estimated from the data by

( ) ( ).

2We w

n m

ai a bi bi i

m n

ed

X X X X

m n

0ill reject if ASL > .H

Inference by Permutation Tests


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Flowchart of The Permutation Test for Mean Shift in One Sample

1 2 1 2 , , ..., , , , ..., n n n n mSample x x x x x x

*11x

*21x

Partition 2 subset B times

(treatment group)

(control group) (treatment group)

(control group)

* * *1 2

ˆ ( ) ( )b b bs s x x

11G 12G

1x 2x11O 12O

1 2ˆ ( ) ( )s s x x

· *ˆ ˆ (#( )) / , and NPerm b nASL B B C

*12x

*22x

21G 22G

*1Bx

*2Bx

1BG 2BG

n m N

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An Example for One Sample Permutation Test by R

http://mason.gmu.edu/~csutton/EandTCh15a.txt

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An Example of Output Results

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1 2 2: ( , x , ..., x )mSample xx

*1x

*1y

Partition subset B times

treatment

subgroup

control

subgroup

11G 12G

1 2 1: ( , y , ..., y )nSample yy

m+n=N

1 2 1ˆ : ( , , ..., , , ..., ) ( , ) ( ) ( )n n n mcombined data d d d d d s s d y x y x

combine

* * *ˆ ( ) ( )b b bs s x y

· *ˆ ˆ (#( )) / , and NPerm b nASL B B C

Flowchart of The Permutation Test for Mean Shift in Two Samples

*2x

*2y

21G 22G

*Bx

*By

1BG 2BG

treatment

subgroup

control

subgroup

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Bootstrap Tests vs. Permutation Tests Very similar results between the

permutation test and the bootstrap test. is the exact probability when . is not an exact probability but is

guaranteed to be accurate as an estimate of the ASL, as the sample size B goes to infinity.

PermASL

BootASL


NnB C

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Cross-validation Methodology R code

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Cross-validation Cross-validation, sometimes called rotation es

timation, is the statistical practice of partitioning a sample of data into subsets such that the analysis is initially performed on a single subset, while the other subset(s) are retained for subsequent use in confirming and validating the initial analysis. The initial subset of data is called the training set. the other subset(s) are called validation or testing s

ets.

http://en.wikipedia.org/wiki/Cross-validation

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Overfitting Problems In statistics, overfitting is fitting a statistical model that has too

many parameters. When the degrees of freedom in parameter selection exceed the i

nformation content of the data, this leads to arbitrariness in the final (fitted) model parameters which reduces or destroys the ability of the model to generalize beyond the fitting data.

The concept of overfitting is important also in machine learning. In both statistics and machine learning, in order to avoid overfitti

ng, it is necessary to use additional techniques (e.g. cross-validation, early stopping, Bayesian priors on parameters or model comparison), that can indicate when further training is not resulting in better generalization.

http://en.wikipedia.org/wiki/Overfitting

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library(bootstrap)

?crossval

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An Example of Cross-validation by R

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output

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Bootstrap Regression Bootstrapping pairs:

Resample from the sample pairs { }. Bootstrapping residuals:

1. Fit by the original sample and obtain the residuals.2. Resample from residuals.

, i ix y

î iy x

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Bootstrapping Pairs by R

http://www.stat.uiuc.edu/~babailey/stat328/lab7.html

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Output

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Bootstrapping Residuals by R

http://www.stat.uiuc.edu/~babailey/stat328/lab7.html

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Bootstrapping residuals

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ANOVA When random errors follow a normal

distribution: When random errors do not follow a

Normal distribution: Bootstrap tests:Permutation tests:

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An Example of ANOVA by R (1) Example

Twenty lambs are randomly assigned to three different diets. The weight gain (in two weeks) is recorded. Is there a difference among the diets?

Reference http://mcs.une.edu.au/~stat261/Bootstrap/

bootstrap.R

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An Example of ANOVA by R (1)

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Output (1)

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Output (2)

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Output (3)

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Output (4)

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Output (5)

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Output (6)

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Output (7)

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The Second Example of ANOVA by R (1)

Data source http://finzi.psych.upenn.edu/R/library/rpart/html/kyp

hosis.html Reference

http://www.stat.umn.edu/geyer/5601/examp/parm.html Kyphosis is a misalignment of the spine. The data are on 8

3 laminectomy (a surgical procedure involving the spine) patients. The predictor variables are age and age^2 (that is, a quadratic function of age), number of vertebrae involved in the surgery and start the vertebra number of the first vertebra involved. The response is presence or absence of kyphosis after the surgery (and perhaps caused by it).

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Output (1)

Data = kyphosis

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Output (2)

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Output (3)

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Output (4)

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Output (5)

#deviance

#p-value

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Output (6)

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Exercises: Write your own programs similar to those

examples presented in this talk.

Write programs for those examples mentioned at the reference web pages.

Write programs for the other examples that you know.

Practice Makes Perfect!81

Documents

1 Nonparametric Methods III Henry Horng-Shing Lu Institute of Statistics National Chiao Tung University [email protected]